Summary: In the Vector Connectivity problem we are given an undirected graph \(G=(V,E)\), a demand function \(\lambda :V\rightarrow \{0,\dots,d\}\), and an integer \(k\). The question is whether there exists a set \(S\) of at most \(k\) vertices such that every vertex \(v\in V{\setminus } S\) has at least \(\lambda (v)\) vertex-disjoint paths to \(S\); this abstractly captures questions about placing servers or warehouses relative to demands. The problem is \(\mathsf {NP}\)-hard already for instances with \(d=4\) [F. Cicalese et al., Theor. Comput. Sci. 591, 60–71 (2015; Zbl 1322.68140)], admits a log-factor approximation [E. Boros et al., Lect. Notes Comput. Sci. 7876, 331–342 (2013; Zbl 1372.68119)], and is fixed-parameter tractable in terms of \(k\) [D. Lokshtanov, unpublished (2014)]. We prove several results regarding kernelization and approximation for Vector Connectivity and the variant Vector \(d\)-Connectivity where the upper bound \(d\) on demands is a fixed constant. For Vector \(d\)-Connectivity we give a factor \(d\)-approximation algorithm and construct a vertex-linear kernelization, that is, an efficient reduction to an equivalent instance with \(f(d)k=O(k)\) vertices. For Vector Connectivity we have a factor \(\mathsf {opt} \)-approximation and we can show that it has no kernelization to size polynomial in \(k\) or even \(k+d\) unless \(\mathsf {NP} \subseteq \mathsf {coNP}/\mathsf {poly}\), which shows that \(f(d){\mathrm {poly}}(k)\) is optimal for Vector \(d\)-Connectivity. Finally, we give a simple randomized fixed-parameter algorithm for Vector Connectivity with respect to \(k\) based on matroid intersection.